Is it true that the order of a nonabelian finite simple group $G$ can be written as the product of the orders of two or more other nonabelian finite simple groups if and only if
$G$ is either an alternating group of degree greater than $8$, or
$G$ is one of the $8$ sporadic simple groups ${\rm J}_2$, ${\rm Suz}$, ${\rm Fi}_{22}$, ${\rm Th}$, ${\rm Fi}_{23}$, ${\rm Co}_1$, ${\rm B}$ or ${\rm M}$?
For example, we have:
- $|{\rm A}_9| = |{\rm A}_6| \cdot |{\rm PSL}(2,8)|$,
- $|{\rm A}_{10}| = |{\rm A}_5|^2 \cdot |{\rm PSL}(2,8)|$,
- $|{\rm A}_{11}| = |{\rm A}_5| \cdot |{\rm PSL}(2,8)| \cdot |{\rm PSL}(2,11)| = |{\rm A}_7| \cdot |{\rm M}_{11}|$,
- $|{\rm A}_{12}| = |{\rm A}_5| \cdot |{\rm PSL}(2,8)| \cdot |{\rm M}_{11}| = |{\rm A}_5| \cdot |{\rm PSL}(2,11)| \cdot |{\rm PSU}(3,3)| = |{\rm A}_7| \cdot |{\rm M}_{12}|$,
- $|{\rm A}_{13}| = |{\rm A}_6| \cdot |{\rm PSL}(2,13)| \cdot |{\rm M}_{11}|$,
- $|{\rm A}_{14}| = |{\rm PSL}(2,7)| \cdot |{\rm A}_6| \cdot |{\rm PSL}(2,11)| \cdot |{\rm PSL}(2,13)|$,
- $|{\rm A}_{15}| = |{\rm A}_6| \cdot |{\rm PSL}(2,11)| \cdot |{\rm PSL}(2,13)| \cdot |{\rm A}_7|$,
- $|{\rm A}_{16}| = |{\rm A}_5| \cdot |{\rm A}_6| \cdot |{\rm PSL}(2,13)| \cdot |{\rm M}_{22}| = |{\rm A}_5| \cdot |{\rm PSL}(2,13)| \cdot |{\rm M}_{11}| \cdot |{\rm A}_8|$ $= |{\rm PSL}(2,7)| \cdot |{\rm A}_6| \cdot |{\rm PSL}(2,11)| \cdot |{\rm PSL}(2,64)| = |{\rm PSL}(2,8)|^2 \cdot |{\rm PSL}(2,11)| \cdot |{\rm PSU}(3,4)|$,
- $|{\rm A}_{17}| = |{\rm PSL}(2,7)| \cdot |{\rm A}_6| \cdot |{\rm PSL}(2,11)| \cdot |{\rm PSL}(2,11)| \cdot |{\rm PSL}(2,16)|$ $= |{\rm PSL}(2,16)| \cdot |{\rm A}_{14}|$,
- $\dots$
- $|{\rm A}_{23}| = |{\rm PSL}(2,11)| \cdot |{\rm PSL}(2,13)| \cdot |{\rm PSL}(2,23)| \cdot |{\rm PSL}(2,49)| \cdot |{\rm J}_3|$,
- $\dots$
and
- $|{\rm J}_2| = |{\rm A}_5|^2 \cdot |{\rm PSL}(2,7)|$,
- $|{\rm Suz}| = |{\rm A}_5| \cdot |{\rm PSL}(2,7)| \cdot |{\rm PSL}(3,3)| \cdot |{\rm M}_{11}|$,
- $|{\rm Fi}_{22}| = |{\rm PSL}(3,3)| \cdot |{\rm M}_{11}| \cdot |{\rm PSp}(6,2)| = |{\rm PSL}(3,3)| \cdot |{\rm PSU}(4,2)| \cdot |{\rm M}_{22}|$,
- $|{\rm Th}| = |{\rm A}_6| \cdot |{\rm PSL}(2,8)| \cdot |{\rm PSL}(2,19)| \cdot |{\rm PSL}(2,27)| \cdot |{\rm PSL}(2,31)| = |{\rm PSL}(2,19)| \cdot |{\rm PSL}(2,27)| \cdot |{\rm PSL}(2,31)| \cdot |{\rm A}_9|$,
- $|{\rm Fi}_{23}| = |{\rm A}_5| \cdot |{\rm PSL}(2,17)| \cdot |{\rm PSL}(2,23)| \cdot |{\rm PSp}(6,3)|$,
- $|{\rm Co}_{1}| = |{\rm A}_5|^2 \cdot |{\rm PSL}(2,13)| \cdot |{\rm PSL}(2,23)| \cdot |{\rm O}^+(8,2)| = \dots $ ($15$ further such products),
- $|{\rm B}| = |{\rm A}_5|^3 \cdot |{\rm PSL}(2,19)| \cdot |{\rm PSL}(2,32)| \cdot |{\rm PSL}(2,47)| \cdot |{\rm F}(4,2)| = \dots $ ($23$ further such products), and
- $|{\rm M}| = |{\rm A}_5|^2 \cdot |{\rm PSL}(2,13)|^3 \cdot |{\rm PSL}(2,17)| \cdot |{\rm PSL}(2,19)| \cdot |{\rm PSL}(2,31)| \cdot |{\rm PSL}(2,41)| \cdot |{\rm PSL}(2,47)| \cdot |{\rm M}_{12}| \cdot |{\rm PSL}(2,59)| \cdot |{\rm PSL}(2,71)| \cdot |{\rm M}_{22}| = \dots$ (many further such products).
In general, the larger the degree $n$, the more such ways to write the order of ${\rm A}_{n}$ seem to exist — although e.g. the expression of ${\rm A}_{23}$ is unique. On the other hand, it seems that no order of a finite simple group of Lie type can be written in that way.
